Integrals

The symbol \(\int f(x)\

The symbol \(\int f(x)\,dx\) represents the entire family of anti derivatives of \(f\). If one anti derivative is \(F(x)\), then the full family is \(

A function \(F\) is an anti

A function \(F\) is an anti derivative of \(f\) if \(F'(x)=f(x)\). Anti derivatives are not unique because adding any constant does not change the der

The constant \(C\) represents any real

The constant \(C\) represents any real number added to an anti derivative. It disappears in definite integrals because it cancels in subtraction at th

The notation \(\int_a^b f(x)\

The notation \(\int_a^b f(x)\,dx\) has a unique value. Here \(a\) is the lower limit and \(b\) is the upper limit. If \(F\) is an anti derivative of \

The area function is \(A(x)=\int_a^x f(x)\

The area function is \(A(x)=\int_a^x f(x)\,dx\). It measures the shaded area from \(a\) to a variable point \(x\) and changes with \(x\).